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 A158810 Coefficients of the differentiated row polynomials of the triangular Hadamard matrices of A158800: p(x,n)=If[n less than or equal to 2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}],If[n greater than m then m+1] 0
 0, -1, 0, -2, -1, -2, 3, 0, 0, 0, -4, -1, 0, 0, -4, 5, 0, -2, 0, -4, 0, 6, -1, -2, 3, -4, 5, 6, -7, 0, 0, 0, 0, 0, 0, 0, -8, -1, 0, 0, 0, 0, 0, 0, -8, 9, 0, -2, 0, 0, 0, 0, 0, -8, 0, 10, -1, -2, 3, 0, 0, 0, 0, -8, 9, 10, -11, 0, 0, 0, -4, 0, 0, 0, -8, 0, 0, 0, 12, -1, 0, 0, -4, 5, 0, 0, -8, 9, 0, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Row sums are: {0, -1, -2, 0, -4, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0,...}. The absolute values of the row sums are: {0, 1, 2, 6, 4, 10, 12, 28, 8, 18, 20, 44, 24, 52, 56, 120,...}. In a quantum Heisenberg matrix mechanics based on the triangular Hadamards where the H(n) behave like wave functions Phi(n), these polynomials are equivalent to the time dependent differentials: Hamiltonian.Phi(n)=-Hbar*I*dPhi(n)/dt LINKS Table of n, a(n) for n=0..89. FORMULA Sum of the k-th row polynomial: p(x,n)=If[n>2^m,Sum[H(2^m)[[k]]*x^(1-k),{k,1,n}]]; t(n,l)=coefficients(p(x,n),x) EXAMPLE {0}, {-1}, {0, -2}, {-1, -2, 3}, {0, 0, 0, -4}, {-1, 0, 0, -4, 5}, {0, -2, 0, -4, 0, 6}, {-1, -2, 3, -4, 5, 6, -7}, { 0, 0, 0, 0, 0, 0, 0, -8}, {-1, 0, 0, 0, 0, 0, 0, -8, 9}, {0, -2, 0, 0, 0, 0, 0, -8, 0, 10}, {-1, -2, 3, 0, 0, 0, 0, -8, 9, 10, -11}, {0, 0, 0, -4, 0, 0, 0, -8, 0, 0, 0, 12}, {-1, 0, 0, -4, 5, 0, 0, -8, 9, 0, 0, 12, -13}, {0, -2, 0, -4, 0, 6, 0, -8, 0, 10, 0, 12, 0, -14}, {-1, -2, 3, -4, 5, 6, -7, -8, 9, 10, -11, 12, -13, -14, 15} MATHEMATICA Clear[HadamardMatrix]; MatrixJoinH[A_, B_] := Transpose[Join[Transpose[A], Transpose[B]]]; KroneckerProduct[M_, N_] := Module[{M1, N1, LM, LN, N2}, M1 = M; N1 = N; LM = Length[M1]; LN = Length[N1]; Do[M1[[i, j]] = M1[[i, j]]N1, {i, 1, LM}, {j, 1, LM}]; Do[M1[[i, 1]] = MatrixJoinH[M1[[i, 1]], M1[[i, j]]], {j, 2, LM}, {i, 1, LM}]; N2 = {}; Do[AppendTo[N2, M1[[i, 1]]], {i, 1, LM}]; N2 = Flatten[N2]; Partition[N2, LM*LN, LM*LN]] HadamardMatrix[2] := {{1, 0}, {1, -1}}; HadamardMatrix[n_] := Module[{m}, m = {{1, 0}, {1, -1}}; KroneckerProduct[m, HadamardMatrix[n/2]]]; M = HadamardMatrix[16]; Table[D[Sum[M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], {n, 1, Length[M]}]; Table[CoefficientList[D[Sum[ M[[n]][[m]]*x^(m - 1), {m, 1, n}], {x, 1}], x], {n, 1, Length[M]}]; Flatten[%] CROSSREFS A158800 Sequence in context: A276990 A127510 A328362 * A129391 A129390 A345255 Adjacent sequences: A158807 A158808 A158809 * A158811 A158812 A158813 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula and Gary W. Adamson, Mar 27 2009 STATUS approved

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Last modified May 20 02:34 EDT 2024. Contains 372703 sequences. (Running on oeis4.)